The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology

Wirtz T, Akemann G, Guhr T, Kieburg M, Wegner RF (2015)
Journal of Physics A: Mathematical and Theoretical 48.

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We consider rectangular random matrices of size $p\times n$ belonging to thereal Wishart-Laguerre ensemble also known as the chiral Gaussian orthogonalensemble. This ensemble appears in many applications like QCD, mesoscopicphysics, and time series analysis. We are particularly interested in thedistribution of the smallest non-zero eigenvalue and the gap probability tofind no eigenvalue in an interval $[0,t]$. While for odd topology $\nu=n-p$explicit closed results are known for finite and infinite matrix size, for even$\nu>2$ only recursive expressions in $p$ are available.The smallest eigenvaluedistribution as well as the gap probability for general even $\nu$ isequivalent to expectation values of characteristic polynomials raised to ahalf-integer. The computation of such averages is done via a combination ofskew-orthogonal polynomials and bosonisation methods. The results are given interms of Pfaffian determinants both at finite $p$ and in the hard edge scalinglimit ($p\to\infty$ and $\nu$ fixed) for an arbitrary even topology $\nu$.Numerical simulations for the correlated Wishart ensemble illustrate theuniversality of our results in this particular limit. These simulations pointto a validity of the hard edge scaling limit beyond the invariant case.
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Wirtz T, Akemann G, Guhr T, Kieburg M, Wegner RF. The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology. Journal of Physics A: Mathematical and Theoretical. 2015;48.
Wirtz, T., Akemann, G., Guhr, T., Kieburg, M., & Wegner, R. F. (2015). The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology. Journal of Physics A: Mathematical and Theoretical, 48.
Wirtz, T., Akemann, G., Guhr, T., Kieburg, M., and Wegner, R. F. (2015). The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology. Journal of Physics A: Mathematical and Theoretical 48.
Wirtz, T., et al., 2015. The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology. Journal of Physics A: Mathematical and Theoretical, 48.
T. Wirtz, et al., “The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology”, Journal of Physics A: Mathematical and Theoretical, vol. 48, 2015.
Wirtz, T., Akemann, G., Guhr, T., Kieburg, M., Wegner, R.F.: The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology. Journal of Physics A: Mathematical and Theoretical. 48, (2015).
Wirtz, Tim, Akemann, Gernot, Guhr, Thomas, Kieburg, Mario, and Wegner, René Florin. “The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology”. Journal of Physics A: Mathematical and Theoretical 48 (2015).
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